What was the question?
Can we find any knot inside certain fractals (in a particular way) ? We studied in particular two fractals: the Menger sponge and the Sierpinski Tetrahedron.
What motivated me?
At that time, I was studying fractals because of a course I had to prepare. I also had fresh in my mind many results of knot theory because I had just given some months before a math academy on knot theory. Moreover, I had just learned about the property of universality of the Menger Sponge. Thus, one day thinking on what the next project could be, it hit me: what if we look for knots inside the Menger Sponge since it is universal.
What did the students do?
This began as a short course on producing iterative fractals and the property of universality. The students were very talented in computing and in crafting, thus they immediately got to program these fractals in such a way that we could explore them visually. That was incredible to me. They also made some origami models of them.
The project was divided into three parts: first, making sure we understood what we were looking for and how to get them. We found easily certain knots in the Menger Sponge, but the tetrahedron was incredibly violent. We could find nothing in there. This first part was exploration, but we had no results.
In the second part, the students settled the question of the Menger Sponge. They proved all knots can be found in there and constructed an algorithm to find them. Furthermore, they created images of them that were beautiful! They were able to find a way to look in the Sierpinski Tetrahedron and finally find the many knots in there. One of the most beautiful trefoils knot I’ve seen was designed like this and then 3D printed by one of them! Finally, we compared the information of the two fractals.
I was so satisfied and impressed by this work that it was decided we would go to the CUMC 2022 to present the work. We had to work an extra month and a half preparing these presentations and posters so that they could work. For this I ask them to present their work to an audience of general grad students and then to two graduate students in knot theory (which made them be extremely nervous!). The whole experience was a success!
Special moments I remember 🙂
Halfway through this project I had my Ph.D. defense. I had to prepare for it and I told the students we would not meet for two weeks. However, I asked them that when we meet again each one of them would have something meaningful to say. I said this a bit sternly more with the intention of making sure they keep working, and not so much that I seriously meant that they had to have results to share.
However, when we met again, they had found all knots in the Menger Sponge and found ways to explore in the Sierpinski Tetrahedron. I was so surprised and proud of them at that moment, not to say extremely impressed with the imaged they were presenting me. I was genuinely shocked.
Outcomes of the work
The students presented a very successful work in the CUMC 2022. Also, at the moment of writing this the work is submitted for publication in a journal and we hope it goes well!
Malors 